The estate had been closed to the public for decades since the safari park in the grounds shut its doors in 1980, but almost 10 miles of new pathways were created as part of a major redevelopment in July 2020.
Four new walks were opened, accessible from a public car park off the A183 at Bournmoor, with families across the region making the most of the surroundings.
The paths have been closed over the winter and were due to reopen at the beginning of February.
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But management have been forced to keep the routes closed as they deal with the aftermath of Storms Arwen, Malik and Corrie, which brought adverse weather conditions across the North East.
Between them, the three weather systems brought down around 750 trees across the estate, blocking roads and footpaths.
Meanwhile, many more have been left in a potentially dangerous condition with some branches hanging by a thread.
Specialist tree surgeons have been brought in to deal with the damage, but the estate says the public will not be allowed back on-site until it is safe to do so.
Estate rural director Johnny Morris said the pathways were a priority and would reopen as soon as possible, but it was too early to say when that would be. Clearing the remainder of the estate would take a considerable time.
He told the Echo: “We are working as quickly as we can and we will reopen is soon as we are safely able to do so.
"It will depend on how the work progresses and, obviously, what happens in the meantime. If we have any more storms arrive, then things may change.
"The clean-up process is going to go on for months and possibly even years.
"A big problem is the sheer volume of what came down, not only in tree numbers but also in size. Immature trees tend to bend in the wind but the bigger ones can’t.”
The situation could have been even more serious if the winds had arrived later in the year, Johnny added.
Estate ranger Ian Nairn explained that staff had still been dealing with the aftermath of November’s Storm Arwen when Storms Malik and Corrie hit earlier in 2022.
Storm Arwen had brought down around 500 trees by itself, with Malik and Corrie adding a further 250 to the tally between them
Efforts to clear the site have also been hampered by the difficulty of finding specialist staff to carry out the work.
The on-site teams have been able to deal with some of it, but contractors have been brought in for other tasks.
Ian continued: "Our first priority was to get the roads open, then we had to look at the houses and other buildings before we could get to the footpaths.”
He added: “We have brought in outside contractors because there is some work we just can’t do ourselves.
"We have been trying to get them in since November, but this is their first week on-site.”